The effect of anisotropic surface energy on the Rayleigh instability

We determine the linear stability of a rod or wire subject to capillary forces arising from an anisotropic surface energy. The rod is assumed to be smooth with a uniform cross-section given by a two-dimensional equilibrium shape. The stability analysis is based on computing the sign of the second variation of the total energy, which is examined by solving an associated eigenvalue problem. The eigenproblem is a coupled pair of second-order ordinary differential equations with periodic coefficients that depend on the second derivatives of the surface energy with respect to orientation variables. We apply the analysis to examples with uniaxial or cubic anisotropy, which illustrate that anisotropic surface energy plays a significant role in establishing the stability of the rod. Both the magnitude and sign of the anisotropy determine whether the contribution stabilizes or destabilizes the system relative to the case of isotropic surface energy, which reproduces the classical Rayleigh instability.